All minerals that crystallize in the orthorhombic, monoclinic, or
triclinic crystal systems are biaxial. Biaxial crystals have 2 optic
axes, and this distinguishes biaxial crystals from uniaxial
crystals. Like uniaxial crystals, biaxial crystals have refractive
indices that vary between two extremes, but also have a unique
intermediate refractive index. Biaxial refractive indices are as
follows:

The smallest refractive index is given the symbol α
(or X).

The intermediate refractive index is given the symbol β
(or Y).

The largest refractive index is given the symbol γ
(or Z)

All biaxial minerals have optical symmetry equivalent to
2/m2/m2/m. But, in each of the crystal systems, the optical
directions have different correspondence to the crystallographic
directions.

In orthorhombic crystals the optical directions correspond to the
crystallographic axes, i.e. the X direction and its corresponding
refractive index, α can be either the a, b,
or c crystallographic axes, the Y direction and β
can be parallel to either a, b, or c, and the Z direction or γ,
can be parallel to either a, b, or c.

In monoclinic crystals, one of the X (α),
Y (β), or Z (γ)
directions or indices is parallel to the b crystallographic axis, and
the other two do not coincide with crystallographic directions.

In triclinic crystals none of the optical directions or indices
coincide with crystallographic directions, although in some rare case
one of the indices might coincide with one of the crystallographic
directions.

The Biaxial Indicatrix

The biaxial indicatrix, like the isotropic and uniaxial
indicatrices, diagrammatically illustrates the refractive index for
vibration directions of light. It is shown in the diagram below.

The biaxial indicatrix has three principle axes, labeled α,
β, and γ.
Directions that have refractive indices between α and
β, are referred to as α'.
Directions with refractive indices between γ and
β are referred to as γ'.
Note that the β direction also must occur in
the plane that includes α and γ.
Similarly, if we were to draw all other possible planes that include the γ
direction, β would have to occur in each of
these as well. This results in two sections that would be circular
with a radius equivalent to the β refractive
index. These two sections are referred to as the circular sections.
In the diagrams below we see the two circular sections, each having a
radius equal to the β refractive index.

In the left-hand diagram some of the other possible planes that include γ
are shown. In the right-hand diagram these planes are removed to
show only the circular sections. Lines drawn perpendicular to the
circular sections are the optic axes. This is why minerals that
exhibit these optical properties are called biaxial.

The acute angle between the optic axes is called the 2V angle.

Just like in uniaxial minerals, if one is looking down one of the optic
axes, light traveling along the optic axis will be vibrating in the β
direction, and thus the mineral would be extinct for all rotation
positions.

The three principal planes of the biaxial indicatrix are shown
here. The plane containing the α and γ
directions also contains the optic axes, which are perpendicular to the β
directions. This plane is called the optic axial plane or
OAP.

The other two principal planes contain the γ
and β directions and the α
and β directions, respectively.

Optic Sign of Biaxial Minerals

The optic sign of biaxial minerals depends on whether the β
refractive index is closer to that of α or to γ. There are several
ways that this can be stated, so we will look at all of them.

Biaxial Positive

A mineral is biaxial positive if β is
closer to α than to γ.

In this case the acute angle, 2V, between the optic axes is bisected
by the γ refractive index direction.
Thus we say that γ is the acute
bisectrix (BXA), because it bisects this angle.

If a table of optical properties of minerals reports the 2V
angle, it usually refers to this acute angle. But some tables
report the 2V as 2Vγ or 2Vα.
Note that in the case of a biaxial positive mineral, 2Vγ
is the acute bisectrix, while 2Vα
bisects the obtuse angle between the optic axes (called the obtuse
bisectrix (BXO).

Biaxial Negative

A mineral is biaxial negative if β is
closer to γ than to α.

In this case the acute angle, 2V, between the optic axes is bisected
by the α refractive index direction.
Thus we say that α is the acute bisectrix.

In the case of a biaxial negative mineral, 2Vα
is the acute bisectrix, while 2Vγ
is the obtuse bisectrix.

Note that 2Vα + 2Vγ
= 180o.

If 2V = 90o the mineral has no optic sign.

If 2V = 0o the mineral is uniaxial.

Optical Orientations of Biaxial Crystals

Just like in uniaxial crystals, we can move the indicatrix around in a
crystal so
long as the indicatrix is kept parallel to the optical directions, and use
this as an aid to determining the optical directions in the crystal.
Several orientations are possible, the most general are shown in the
diagram below. Note that in this diagram we have not shown the
crystallographic axes, because different minerals will show different
relationships between optical directions and crystallographic directions
as discussed above.

If a crystal oriented on the microscope stage with its α
vibration direction perpendicular to the stage, then the β
and γ vibration directions will be the two
privileged directions in the crystal, as for the face labeled A in
the diagram below. With such a face it would be possible to
determine the β and γ
refractive indices by using the Becke line method and various oils
when either of these privileged directions is oriented parallel to
the polarizer. Just as in uniaxial minerals, the crystal would
be extinct with the analyzer inserted when either of the privileged
directions are parallel to the polarizer. At any other
orientation of the two principal vibration directions the crystal
would exhibit an interference color that reflects the thickness of the
crystal and the birefringence for this orientation (γ
- β). Note that this would not be the
maximum interference color for this crystal.

If a crystal face such as B in the diagram above is parallel to the
microscope stage, one would be looking down the β
direction in the crystal. In this orientation the optic axial
plane (OAP) would be parallel to the stage, and the α
and γ refractive indices could be
determined using the Becke line method and various oils. Because
the β direction in this orientation is perpendicular to or normal to
the OAP, the β direction is often referred to
as the optic normal direction (O.N.).

In this orientation the interference color exhibited by positions off
of extinction would again reflect the thickness of the crystal and the
birefringence. But the birefringence for this orientation would
be the absolute or maximum birefringence possible for this crystal (γ
- α).

If a face like C were lying parallel to the stage, it would show
another of the principal planes of the indicatrix, this time showing
the α and β
vibration directions of the crystal. Again, the refractive
indices corresponding the α and β
could be determined in this orientation. Note that for crystal
faces A or C, we would be looking down either the BXA or BXO
depending on the optic sign of the mineral. The interference
color seen with the analyzer inserted will reflect an intermediate
birefringence for the crystal (β - α).

If a face like D were lying parallel to the stage, it would show the
circular section and one of the optic axes would be orientated
perpendicular to the stage. For this orientation the crystal
would show no change in relief on rotation of the stage with the
analyzer not inserted, and would remain extinct through a 360o
rotation with crossed polarizers. For any rotation position the β
refractive index could be determined.

Faces like E and F, if lying parallel to the stage, would have privileged
directions corresponding to random vibration directions. α',
γ' and β could
be measured, but this would be of little use because all of these
refractive indices are intermediate.

While knowledge of the optical directions can provide a means to
measure the principle refractive indices, this is not often done
because much of the information necessary to identify a biaxial mineral
can be obtained from interference figures.

Biaxial Interference Figures

Four primary types of biaxial interference are seen.
Only two of these are commonly used, but it is essential to discuss all four
so that you can recognize each.

Acute Bisectrix Figure (BXA)

Looking down the acute bisectrix, (the γ
direction perpendicular to the stage if the crystal is optically positive,
or the α direction perpendicular to the stage
if the crystal is negative), at 45o off extinction in conoscope
mode, one would see the interference figure shown in the left-hand diagram
below.

The dark isogyres mark the positions where light vibrating parallel
to the polarizer has passed through the crystal.

At the points of maximum curvature of the isogyres are the two
melatopes that mark the positions where rays that traveled along the
optic axis emerge from the field of view.

Note that the distance between the two melatopes is proportional to
the angle 2V between the optic axes.

Also seen are isochromes, which show increasing interference colors
in all directions away from the melatopes. The number of isochromes
and maximum order of the interference colors seen will increase with
increasing thickness and absolute birefringence of the crystal.

Shown in the figure is the trace of the optic axial plane which
includes the two optic axes.

As the stage is rotated 45o from this initial position, the
isogyres will close to produce a cross. In this position the crystal
would be extinct in orthoscope mode. The melatopes will be
rotated so that both lie along the N-S cross hair.

Rotation by an additional 45o will result in the isogyres
then separating again to show the interference figure in the third
diagram. Another 45o rotation will again cause the
isogyres to close into a cross, this time with the OAP lying parallel to
the polarizing direction of the microscope. The crystal would again
be extinct in orthoscope mode. Another 45o rotation would
return the view to the first diagram in the series.

Optic Axis Figure (OA)

If one of the optic axes is oriented perpendicular to the stage,
placing the microscope in conoscope mode will result in an
optic axis interference figure. This is similar to the BXA
figure, except one of the isogyres and melatopes will be outside of the
field of view (unless the 2V angle is very small).

During rotation of the stage, the melatope will remain at the cross-hair
intersection and the isogyres will close to form an off-centered cross and
then separate to show the curved isogyre in the adjacent quadrant of the
field of view.

OA figures are easiest to find among randomly oriented grains, because
a grain that shows such a figure will show no change in relief on a 360o
rotation (analyzer out), and will remain extinct through a 360o
rotation (analyzer inserted).

Obtuse Bisectrix Figure (BXO)

A BXO figure will be similar to the BXA figure,
except that the melatopes will be outside of the field of view most of the
time during a 360o rotation. Still, every 90o
the broad cross will form as the OAP becomes parallel to one of the cross
hairs.

Optic Normal Figure (O.N.)

If the principal β direction of the indicatrix is oriented
perpendicular to the stage, such that the crystal's privileged directions
are α and γ, then
changing to conoscope mode will produce an optic normal figure, also
called a flash figure. In this figure, when one of the two
privileged directions lines up with the polarizer, a broad cross covering
almost the entire field of view will be observed. This cross,
however will quickly disappear after just a slight rotation of the stage
(this is why it is often called a flash figure). If one sees an
optic normal figure, then the interference colors observed in orthoscope
mode will reflect the absolute or maximum birefringence of the mineral, as
discussed above.

Determination of Optic Sign

Biaxial interference figures are most useful for the
determination of optic sign and estimation of the 2V angle, both of which
are useful diagnostic properties of biaxial minerals. The two
most useful are the BXA figure and OA figure.

Acute Bisectrix Figure

To determine the optic sign of a biaxial mineral from a BXA
figure, position the isogyres so that the melatopes are in the NE and SW
quadrants. There should be an area near the melatopes that shows a 1o
gray interference color. Observe this area as you insert the 550nm
or 1o red compensator. If the 1o gray area in
region between the two isogyres turns yellow, the mineral is biaxial
positive. If the 1o gray area inside of both the isogyres
turns yellow the mineral is biaxial negative. Note that it is
easiest to remember this if you imagine the area inside the isogyres turns
yellow and a line drawn inside the isogyres crosses the slow direction of
the compensator like the vertical stroke on a plus sign. For a
negative mineral, the line connecting the two yellow areas is parallel to
the slow direction in the compensator, forming the minus sign.

Centered Optic Axis FigureOptic axis figures probably provide the easiest method for
determination of optic sign because grains with an orientation that would
produce an OA figure are perhaps the easiest to find.

The method is similar to the BXA figure, except you will be
looking at only one of the isogyres. Again place the isogyre so that
the inside of the isogyre is in the NW quadrant. Find the area that
shows 1o gray close to the melatope. Observe this area as
the 550 nm compensator plate is inserted. If the area outside of the
isogyre turns yellow, the mineral is biaxial positive. If the area
inside the isogyre turns yellow, the mineral is biaxial negative.

Off-centered FiguresProbably even easier to locate are off-centered OA or BXA
interference figures.

The method for optic sign determination in off-centered figures is
essentially the same as for BXA and OA figures. Position
the isogyres so that it fits best in either the NE or SW quadrant.
Observe the gray area near the melatopes and note the color change on
insertion of the 550 nm compensator. If the gray area outside the
isogyre turns yellow, the mineral is biaxial positive. If the gray
area outside isogyre turns blue and the gray area inside the isogyre turns
yellow, the mineral is biaxial negative.

How to Locate Different types of Biaxial Interference Figures

The best indicator of the type of interference figure a given grain
will produce is the level of interference colors exhibited by the grain in
orthoscope mode. Note that the O.N. figure will occur on
grains that show the maximum interference colors. Such a grain will
give the best indicator of the absolute birefringence of the
mineral. OA. figures and Off-centered OA figures will be easiest to
locate because the grain will either be completely extinct on a 360o
rotation (OA figure) or will show very low birefringence (off-centered
OA).

Type of Interference Figure

Level of Interference Colors

O.N.

Maximum

BXO

Relatively High

BXA

Relatively Low

O.A.

None

Off-centered O.A.

Very Low

How to Distinguish Biaxial from Uniaxial Interference Figures

Biaxial minerals can often be distinguished from uniaxial minerals on
the basis of an interference figure.

To do so, rotate the stage until the isogyre rests on intersection of
the cross hairs. If there is any curvature to the isogyre, the
mineral is biaxial. If the isogyre is straight, then the mineral is
either biaxial with a low 2V or is uniaxial. In the latter case
further tests will have to made on other grains to make the distinction.

Estimation of 2V

Precise determination of 2V can only be made by
determining the 3 principal refractive indices of the mineral. But,
2V can be estimated from Acute Bisectrix figures and Optic Axis figures
using the diagrams shown here.

Acute Bisectrix Figure

Recall that for a BXA figure the distance between the
melatopes is proportional to the 2V angle. To estimate the 2V from a
BXA figure, one first needs to know the numerical aperture (N.A.)
of the objective lens used to observe the interference figure. The
microscopes in our labs have an N.A. of 0.65, while research microscopes
generally have a higher N.A. of 0.85. The diagram shown here gives a
visual estimate of the 2V angle for objective lenses with these two values
of N.A. for a mineral with a β refractive index
of 1.6.

Remember that if the 2V is 0o the mineral is uniaxial, and
would thus show the uniaxial interference figure. The separation of
the isogyres or melatopes increases with 2V and the isogyres eventually go
outside of the field of view for a 2V of 50o with the smaller
N.A., and about 60o for the larger N.A.

Since the maximum 2V that can be observed for a BXA figure
depends on the β refractive index, the chart
shown here may be useful to obtain more precise estimates if the β
refractive index is known or can be measured.

Optic Axis Figure

2V estimates can be made on an optic axis figure by noting the
curvature of the isogyres and referring to the diagram shown here. Note
that the curvature is most for low values of 2V and decreases to where the
isogyre essentially forms a straight line across the field of view for a
2V of 90o. For a 2V of 0o the mineral is
uniaxial and the isogyres form a cross with straight isogyres.

Extinction Angle

Extinction angle is the property that involves determining
the angle between the a crystallographic direction as exhibited by a
crystal face or cleavage and one of the principal vibration
directions. We have already discussed parallel and symmetrical
extinction in uniaxial minerals. The concept is the same for Biaxial
minerals, except that the extinction angles could be different from 90o
or 0o, as is the case for parallel extinction.

Three
different cases are observed depending on whether the mineral is
orthorhombic, monoclinic, or triclinic.

Orthorhombic Minerals

In orthorhombic minerals the principal vibration directions are
coincident with the 3 crystallographic axes. Thus, for most
orientations of the mineral on the stage, cleavages that are parallel to a
crystallographic axis will show extinction that is parallel to or at 90o
to such a cleavage.

Shown here on the left is an orthorhombic mineral with {010} cleavage. Since
this cleavage is parallel to the a and c crystallographic axes, and since
the α, β, and γ
vibration directions are also parallel to the a, b, and c axes, this
mineral will show parallel extinction with respect to the {010} cleavage.

On the right is an orthorhombic mineral with {110} cleavage.
Again, since the cleavage is parallel to the c crystallographic axis, and
one of the principal vibration directions is also parallel to the c axis,
the mineral will show parallel extinction if the mineral is lying on any
face parallel to the c axis [i.e. (010), (110)] or any other random face
such as (210), (130), etc. (not shown in the diagram).

If the crystal is lying on a face that is not parallel to the c axis,
it will show symmetrical extinction with respect to the cleavages.

Monoclinic Minerals

In monoclinic crystals, only one of the principal vibration directions
will coincide with a crystallographic axis. The others will be at
some angle to the crystallographic axes.

In the case shown here, a monoclinic mineral with {110} cleavage will
show extinction at some angle other than 0o or 90o
to the cleavage direction. This angle will vary from some maximum angle
to 0o, depending on which crystal face is parallel to the
stage. In the case shown here, the maximum angle will be seen if one
is looking down the b axis or β direction,
that is the face (010) [or (00)].
Only if the (100) face is parallel to the stage will parallel
extinction be observed.

This is important, because if a mineral is reported to have inclined
extinction with respect to a cleavage, note that it is possible, given the
right orientation of the grain, for the mineral to exhibit parallel
extinction.

For minerals that show inclined extinction, tables of determinative
properties usually list the maximum extinction angle in the form γ
∠ c or Z∠ c.

Triclinic Minerals

In triclinic minerals none of the principal vibration directions are
constrained to coincide with crystallographic directions. Thus, asymmetrical
and inclined extinction are to be expected. Again, in tables of
determinative properties of minerals the maximum extinction angle will be
given in a form such as γ∠ c or Z∠ c. Also, note that there will be some
orientation of the grain where the extinction will be parallel, so
extinction angle should be tested on several grains of the same mineral.

Sign of Elongation

For minerals that commonly show an elongated habit, the
sign of elongation could be an important property. In biaxial
minerals sign of elongation is only important if the mineral tends to be
elongated in the direction of either γ or α,
since these are the maximum and minimum refractive indices. Sign of
elongation is determined by positioning the grain so that its elongation
direction is parallel to the slow direction of the compensator.
Before inserting the compensator find an area near the edge of the grain
that shows a 1o gray interference color. If the gray area
turns blue then the mineral's slow direction (γ
direction) is parallel to the slow direction in the compensator. If
the gray areas turns yellow, then the mineral's fast direction (α
direction) is parallel to the slow direction in the compensator.

Note that sign of elongation is usually not reported for minerals that
are elongated in the β direction, since some
faces would show length fast, and other faces would show length
slow.

Pleochroism

As discussed under uniaxial minerals, pleochroism is the
property where the mineral shows a different absorption color associated
with different vibration directions. In uniaxial minerals the two
main vibration directions could have different absorption colors, and any
intermediate direction would show an intermediate color. In biaxial
crystals, three different absorption colors are possible, one associated
with each of the principal indices. Intermediate directions will
give intermediate colors.

The pleochroic
formula is usually given in terms of the three principal refractive
indices, for example a biaxial mineral could have the pleochroic formula α
= red, β = pink, γ =
clear.

Of course, since only two vibration directions can
be observed at any one time, only two of the colors will be seen on
rotation of a given grain. Thus, several grains of the same mineral
must be observed in order to determine the pleochroic formula of the
mineral.

Other Observations

We here discuss some other properties that may be
exhibited by some minerals. If any of these occur, they may be
useful diagnostic features of the mineral.

Zoning

Zoning occurs as a result of incomplete reaction of solid solutions and
results in the chemical composition of the mineral changing through the
mineral. Optical properties, depend on chemical composition,
and thus if the composition changes through a crystal, the optical
properties will vary through the crystal as well. In particular, the
orientation of the principal vibration directions may change, and thus the
angle at which the mineral goes extinct may change. This can be
observed by rotating a zoned crystal and noting that the whole crystal
does not go extinct all at once. Each part that goes extinct at the
same time or has the same interference color at the same time has a chemical composition distinct from other parts of the same
crystal.

Twinning

Since twinning is an intergrowth of two or more crystals, optical
properties will change at the boundaries between twins.

Thus, different parts of the crystal will go extinct at different times
as a result of twin planes. Plagioclase polysynthetic twinning is
seen as dark and light colored stripes running through the crystal under
crossed polars (left-hand illustration). Cyclical twins and simple
contact twins are shown in the other illustrations.

Exsolution Lamellae

Some minerals that form solid solutions at high temperature exsolve as
they pass through lower temperatures. This exsolution often results
in domains of one mineral inside of the other, called exsolution
lamellae. This is very common in the alkali feldspars that occur in
plutonic igneous rocks, as shown here. It also occurs in other
minerals, particularly the pyroxenes. When exsolution lamellae are
present, they can be very diagnostic of the mineral.

Undulatory Extinction

Deformation of minerals can result in strain of the crystal structure,
which causes different parts of the same mineral to have different
crystallographic and therefore optical orientations. When this
occurs, the parts of the crystal with different orientations will go
extinct at different rotational positions. This is referred to as
undulatory extinction. It is common in quartz found in metamorphic
rocks.

Abnormal Interference Colors

As discussed under uniaxial minerals, if a mineral has strong absorption of certain wavelengths
of light, these same wavelengths will be absorbed by the crystal with the
analyzer inserted, and thus the crystal may produce an abnormal or anomalous
interference color, one that is not shown in the interference color
chart. For example, imagine a crystal that shows strong absorption
of all wavelengths of light except green. Thus, all other
wavelengths are absorbed in the crystal and the only wavelengths present
that can reach the analyzer are green. The crystal will thus show a
green interference color that is not effected by the other wavelengths of
light, and thus this green color will not appear in the interference color
chart. When a mineral exhibits abnormal interference colors, it will
usually be listed as one of the diagnostic properties.

Associations

Some minerals commonly occur with other minerals in the
same rock due to the chemical composition of the rock. Likewise,
some minerals do not occur in association with other minerals.
Mineral associations can be very useful diagnostic properties. For
example, nepheline and quartz do not usually occur with one another, nor
does Mg-rich olivine and quartz. Thus, if you see quartz in a rock,
you will not likely find Mg-rich olivine or nepheline. Aluminous schists,
result from metamorphism of shales, which contain an abundance of Al-rich
clay minerals. So, we expect to find Al-rich minerals in aluminous
schists, like garnet, muscovite, alkali feldspar, biotite, and an Al2SiO5
mineral like kyanite, andalusite, or sillimanite.

Mineral associations often make mineral identification
much easier because you know what minerals to expect. These will be
discussed to a much greater extent in EENS 2120, Petrology. In
your Deer, Howie and Zussman book, mineral associations are referred to as
Paragenesis.

Summary of Optical Properties

Determination of Isotropic, Uniaxial, or Biaxial
Character

Mineral is

Isotropic if all grains are extinct under crossed
polars during 360o rotation.

Uniaxial if it gives a uniaxial interference
figure.

Biaxial if it gives a biaxial interference figure.

Estimation of Birefringence - in thin section
with thickness of minerals of 0.03 mm, birefringence is estimated
using interference color chart. Note that only absolute birefringence
is diagnostic:

|ω - ε| for uniaxial minerals

(γ - α) for
biaxial minerals

Optic sign -

from uniaxial centered or near centered
interference figure for uniaxial minerals.

from biaxial BXA or OA centered or near
centered interference figure.

Relief - from comparison with surrounding
minerals or cement in which the crystals are mounted, or with oil in immersion method.

Absorption color - if present, may be observable
in isotropic, uniaxial, and biaxial minerals with analyzer not
inserted.

Uniaxial minerals may have pleochroic formula: ω
= color1, ε = color2. If optic
axis is perpendicular to the stage, only one color will be
observed.

Biaxial minerals may have pleochroic formula
α= color1, β
= color2, γ = color3, but only 2 colors
will be observed in any one grain, unless the optic axis is
perpendicular to stage - then only one color.

Sign of Elongation

For uniaxial minerals that are elongated parallel to
the C axis, sign of elongation will + or slow if the mineral is
optically positive, and - or fast if the mineral is uniaxial
negative. For minerals that do not show an elongated habit
this property is not reported.

For biaxial minerals that are elongated parallel
to γ, the sign of elongation will be +
or length slow. If they show a habit elongated parallel to α,
the sign of elongation will be - or length fast. If the mineral
does not commonly show an elongated habit or if it is elongated
parallel to β, the sign of elongation
will not be reported.

Extinction Angle

Uniaxial minerals that have cleavage parallel to
the c axis will have parallel extinction on all faces lying with
the c axis parallel to the stage, and have symmetrical extinction
on all other faces if the cleavages intersect.

Biaxial minerals that have cleavage parallel to
the c axis will usually have inclined and asymmetrical extinction
if the mineral is triclinic or monoclinic, and will have parallel
and symmetrical extinction if the mineral is orthorhombic.
Extinction angles that reported will be the maximum possible acute
angle, and thus different orientations of the minerals could
produce extinction angles from 0 to 45o.

Other Properties, as discussed above

zoning

twinning

exsolution lamellae

undulatory extinction

abnormal interference colors

Examples of questions on this material that could be asked on an exam

What is the correspondence between the three principal vibration directions and the crystallographic axes in biaxial crystals?

How is the optic sign of biaxial crystals defined? Give two different ways.

Draw a centered BXA figure for a mineral that is highly birefringent with a positive 2V = 30o. Show the isogyres in the northeast and southwest quadrants. On the diagram label the following: (a) melatope(s), (b) isogyres, (c) isochromes, and (d) the 2V. Indicate the effect of inserting the first order red compensator, by showing the compensator and its slow direction along with any color changes that might occur in the interference figure.

For a biaxial or uniaxial mineral give as many methods as you can for locating a grain in which the optic axis will be oriented perpendicular to or nearly perpendicular to the microscope stage.